Maximal symplectic systolic constant for convex domains (including Lagrangian products)

Determine the value of the maximal symplectic systolic constant Sys(K^{2n}) for the class K^{2n} of convex domains in R^{2n}, defined by Sys(K^{2n}) := sup_{K ∈ K^{2n}} c_EHZ(K) / ((n! Vol(K))^{1/n}), where c_EHZ denotes the Ekeland–Hofer–Zehnder capacity and Vol denotes Euclidean volume. In particular, determine this value for the subclass of Lagrangian products K × T ⊂ R_q^n × R_p^n of convex bodies, even in the case where one of the factors is the Euclidean ball.

Background

The paper introduces the maximal symplectic systolic constant Sys(K^{2n}) as the supremum of the normalized Ekeland–Hofer–Zehnder (EHZ) capacity over all convex domains in R^{2n}. For convex domains, Sys(K^{2n}) is known to be bounded above by a universal (but inexplicit) constant, improving earlier dimension-dependent bounds. The ratio is unbounded for the broader class of star-shaped domains.

The authors’ counterexample to Viterbo’s conjecture motivates determining the exact value of Sys(K^{2n}). They highlight the subclass of Lagrangian product domains K × T ⊂ R_q^n × R_p^n, noting that even when one of the factors is the Euclidean ball, the problem of determining the value remains open.